accelerator division - lunds universitet...delay/us hbw/hz critical gain hbw/hz critical gain 1 518...

!!!!! ! ! !!!!!!!!!!

!

!!!Accelerator Division !!!!!!!!!!

Rihua Zeng, Anders J. Johansson, Karin Rathsman and Stephen Molloy

Influence of the Droop and Ripple of Modulator

on Klystron Output

23 February 2012

ESS AD Technical Note ESS/AD/0033

!

Influence of the Droop and Ripple of Modulator

on Klystron Output

R. Zeng, A. J. Johansson, K. Rathsman, S. Molloy

September 15, 2011

Abstract

The variations of the phase and amplitude of klystron output due tothe change in klystron cathode voltage is investigated in this note. Themechanism and the e↵ectiveness of the feedback control to suppress thevariations are given. To understand the limitation of the feedback, bothproportional controller and proportional-integral controller used in feed-back loop are simulated and analyzed respectively for superconductingcavity and normal conducting cavity. The tolerances of the droop andripple in cathode voltage are shown according to the data and resultsobtained. All the simulations and calculations are performed with MAT-LAB. The data and results are listed in detail so as to enable comparisonwith further studies and measurements at ESS.

1 Introduction

In accelerator, the klystron su↵ers the droop and ripple e↵ect resultingfrom the modulator (klystron cathode voltage supplier), while the droopand ripple in klystron cathode voltage leads to a phase and amplitudemodulation on klystron output. At ESS, there might be potentially seriousdroop and ripple because of long pulse up to 3 ms. It is important for usto know to what extent the droop and ripple a↵ects the klystron output,and how much we can tolerate.

2 Phase and Amplitude Variations

To find how the klystron output is a↵ected, consider a simplified two-cavity klystron as shown in Figure 1.

Electron beams generated out from Electron gun are firstly acceler-ated by the cathode voltage V and then modulated by the RF signal atinput cavity. After passing through the drift space of the length L, thebeams finally induce the required RF signal at output cavity. The RFoutput phase varies if there is any change in the time to travel throughthe drift space, which is highly a↵ected by the change in cathode voltage.The phase variation due to the voltage change can be calculated by thefollowing equations:

1

Figure 1: Two-cavity klystron schematic diagram [1]

t = L

�= L�

eV

m

, (1)

✓ = !t, (2)

d✓ = − !L�2eVm

dV

V, (3)

where v is the velocity of the beam, V is the cathode voltage of klystron,L is the length between the RF input and output of klystron, t is thetime delay of the beam for a L length travelling, ✓ is the correspondingphase delay with RF frequency !, d✓ is the phase variation, dV is thevoltage variation and e�m is the charge to mass ratio of an electron. Moreaccurate result can be calculated by the relations given in Appendix A [2]in consideration of relativistic e↵ects.

There is also variation in amplitude as a result of the change in klystroncathode voltage. The calculation of the amplitude variation [3] is basedon the following relations:

Pout

∝ V 5�2, (4)

Vout

∝ P1�2out

, (5)

Vout

∝ V 5�4, (6)

dVout

Vout

= 5

4

dV

V. (7)

Assuming that the RF input is cos(!t) and there is error �V in cath-ode voltage, the RF output signal is therefore modulated by �✓ in phaseand �A in amplitude, which can be written as:

Vout

= (A +�A)cos(!t +�✓), (8)

where A is the klystron gain. For example, considering a klystron withL = 2.5m,f = 508MHz, and the cathode voltage = 80kV , a cathode volt-age change �V �V = 1% results in a phase variation �✓ of 13° (12° for

2

relativistic case), and amplitude variation �A�A of 1.25%. Some mea-surements for the phase and amplitude variations in other labs are listedin Table 1. They are in agreement with the ones calculated above.

Table 1: Measurement for the phase and amplitude variations in other labs

RF frequency Cathode Phase variation Amplitude

/MHz voltage change /deg. variation

JPARC[4] 312 3.40% 25 ˜8%(power)

SNS[5, 6] 805 3% ˜50(max) ˜8%(power)

PEPII[7] 476 ˜14° /kVSACLAY[8] 704.4 200V@95kV 10° /kV@92kV

It appears that the phase is much more influenced by the ripple thanthe amplitude. Therefore, we will mainly discuss the phase variations inlater sections. If there is ripple component �V sin(!t) mixed in cathodevoltage, the RF output signal correspondingly become:

Acos(!t + k�V sin(!t)). (9)

It is the common expression of the frequency modulated signal, whichhas been well studied in radio communication. Because the droop canbe also viewed as the ripple with DC component or very low frequencycomponent, we will therefore discuss just the case of the ripple componentswith various frequencies in the following sections.

3 Suppression of variations

The amplitude and phase variations of klystron output resulted from mod-ulator droop and ripple can be suppressed in feedback loop by a factorof G+1, where G is the loop gain. However, the feedback gain G cannotbe unlimitedly increased due to the loop delay. The low gain limits thefeedback performance and leaves steady errors. The integral gain is thenintroduced to eliminate the steady error and also the low frequency noises,but have a poor performance at high frequency. More details will be givenin following sections.

3.1 Loop delay and loop gain

The block diagram of a simplified feedback loop for cavity phase andamplitude (or I & Q) control is given in Figure 2, where the klystron,cavity and detector are all simulated as the one-order low pass filters[9, 10], and only proportional controller is considered. The loop delayis generally of the order of µs in LLRF system, which is the key factorcausing loop instability and limiting loop performance. If we note thatthe 3-dB cut-o↵ frequencies of the klystron and the detector are usuallymuch higher than the Cavity, the block diagram can be further simplifiedinto Figure 3.

3

Figure 2: Block diagram of simplified the feedback control loop

Figure 3: Block diagram of further simplified feedback control loop

4

The cavity behaviours like a low pass filter to phase and amplitudein the simplified control loop without considering Lorentz detuning andsynchronous phase operation. Its transfer function can be written as [9]:

Hcav

(f) = fhbw

jf + fhbw

, (10)

where f is the frequency variable, and fhbw

is the cavity half bandwidth .The open loop transfer function H

o

(f) of the control loop in Figure 3has the expressions as shown in Equations 11, 12, and 13, assuming thatthe bandwidths of the klystron and LLRF detector are both larger enoughthan the cavity bandwidth.

Ho

(f) = GHcav

e−j2⇡⌧f , (11)

�Ho

(f)� = G�� f

fhbw�2 + 1

, (12)

' =∠Ho

(f) = −arctan� f

fhbw

� − 2⇡⌧f. (13)

When f � fhbw

, the last two equations can be approximately written as:

�Ho

(f)� = Gfhbw

f, (14)

' = −⇡2− 2⇡⌧f. (15)

Instability of the closed loop can be concluded from the characteristicsof open loop transfer function, which occurs when the magnitude and thephase angle of the open loop transfer function are under special conditionsas follows: �H

o

(f)� ≥ 1, (16)

' = −⇡ + n ⋅ 2⇡, n = 0,±1,±2, ... (17)

Combining the Equation 14, 15, 16 and 17, we can calculate the criticalfrequency where the phase equals −180°(−⇡) and the critical loop gainwhere the magnitude equals 0 dB. The lower the delay is, the higher thecritical loop gain will be. Consequently the better feedback performancecould be achieved. The critical loop gains obtained under di↵erent loopdelays are listed in Table 2 both for superconducting cavity (with halfbandwidth of 518 Hz, the current design value for ESS high beta cavity)and normal conducting cavity (assuming half bandwidth of 10 kHz).

The loop delay ⌧ is set to 2µs in the following sections, which is thereasonable value considering the complicated perturbation factors (ripple,Lorentz detuning, passband modes, etc.), even though 1µs loop delay isachieved in some cavities at SNS (but some cavities are still larger than1µs, even up to 1.5µs). At 2 µs loop delay, the critical gains are 241 and12 for superconducting cavity and normal conducting cavity respectively,and the critical frequency is 125kHz for both.

It is at risk to have loop gain below but close to the critical gain,which causes overshoot as shown in Figures 4 and 5. We can see clearlyfrom the figures that the loop gain around 0.3G (G is the critical gain)

5

Table 2: Critical loop gains obtained under di↵erent loop delays

Superconducting cavity Normal conducting cavity

Delay/us HBW/Hz Critical gain HBW/Hz Critical gain

1 518 482 10000 25

1.5 518 321 10000 16

2 518 241 10000 12

2.5 518 193 10000 10

3 518 160 10000 8

is smooth enough without overshoot, which means the loop gain of 72 forsuperconducting cavity and 4 for normal conducting cavity. In practice,at SNS the average loop gain is about 50 for superconducting cavity and6 for normal conducting cavity [6, 11]. In JPRAC where only the normalconducting cavities are used the average loop gain is about 5 [12]. InFLASH, the loop gain is about 70 ∼ 100 [9, 13].

3.2 Suppression of the klystron output variation

by proportional gain

The influence of the ripple and droop of the modulator on klystron isequivalent to adding a noise to the klystron output. The block diagramof the feedback loop with noise is given in Figure 6, which is applicablefor both phase and amplitude loops (or I &Q loops). The noise inside thecavity bandwidth is fully passed to the cavity in open loop via the pathfrom r to y, while suppressed by a factor of G+1 (G is the loop gain) inclosed loop, as shown in Figure 7.

The larger the loop gain, the better the loop performance against noise.The characteristics are shown obviously in Figures 8 and 9. However, it isalso found that larger gain induces bigger overshoot and longer oscillation.

With only the proportional controller, it is hard for the normal con-ducting cavity to deal with large perturbation (with a loop gain of 6, forexample, a perturbation of 15° phase variation can be only suppressedto 2.5°). What’s worse, it leaves big steady error as seen from Figure 4.These are the reasons why the integral controller needs to be introducedas well in the feedback control.

3.3 Suppression of the klystron output noise by

proportional-integral controller

As mentioned above, to eliminate the steady error and better suppressthe noise, apart from the proportional controller we also have to employthe integral controller. The larger the integral gain is, the better thenoise suppression performance of the integral controller will be at lowfrequency. However, larger integral gain consumes more phase margin,thereby increasing the risk of raising the instability.

6

0 0.5 1 1.5 2 2.5 3 3.5 4

x 10!5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Step Response

Time (sec)

Am

plitu

de

0.7G0.6G0.5G0.4G0.3G

Figure 4: The step response of feedback for normal conducting cavity(G=12)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 10!5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Step Response

Time (sec)

Am

pli

tud

e

0.7G0.6G0.5G0.4G0.3G

Figure 5: The step response of feedback for superconducting cavity(G=241)

7

Figure 6: Block diagram of the proportional control feedback loop with noise

102

103

104

105

106

!70

!60

!50

!40

!30

!20

!10

0

Mag

nit

ud

e (

dB

)

Bode Diagram

Frequency (Hz)

r to y open loop

r to y closed loop

Figure 7: Noise suppression performance of the open loop and closed loop forproportional feedback control (Loop gain: G=72)

8

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 10!5

0

0.005

0.01

0.015

0.02

0.025

Step Response

Time (sec)

Am

plitu

de

0.7G0.6G0.5G0.4G0.3G0.2G

Bode Diagram

Frequency (Hz)10

310

410

510

610

710

8!100

!90

!80

!70

!60

!50

!40

!30

Mag

nit

ud

e (

dB

)

0.7G0.6G0.5G0.4G0.3G0.2G

Figure 8: Closed loop noise suppression for superconducting cavity(G=241)

0 0.5 1 1.5 2 2.5 3 3.5 4

x 10!5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Step Response

Time (sec)

Am

pli

tud

e

0.7G

0.6G

0.5G

0.4G

0.3G

0.2G

103

104

105

106

107

108

!80

!70

!60

!50

!40

!30

!20

!10

Ma

gn

itu

de

(d

B)

Bode Diagram

Frequency (Hz)

0.7G0.6G0.5G0.4G0.3G0.2G

Figure 9: Closed loop noise suppression for normal conducting cavity(G=12)

The block diagram of the feedback loop with PI (proportional-integral)controller [14] is given in Figure 10, where Kp is the proportional gain andKi is the integral gain. In the feedback loop with PI controller, the openloop transfer function can be written as:

Ho

(f) =Kp

�1 + Ki

j2⇡f�H

cav

(f) e−j2⇡⌧f

=Kp

� j2⇡f +Ki

j2⇡f�� f

hbw

jf + fhbw

� e−j⇡⌧f ,

(18)

�Ho

(f)� =Kp

��1 + � Ki

2⇡f�2��1 + � f

fhbw

�2, (19)

' =∠Ho

(f) = arctan�2⇡fK

i

� − arctan� f

fhbw

� − ⇡

2− 2⇡⌧f. (20)

We firstly consider the case without loop delay (⌧ is 0). It is obviously seen

from Equation 20 that when arctan � 2⇡f

Ki�>arctan � f

fhbw�, i.e., K

i

<fhbw

,

the loop phase is always larger than −90° (−⇡�2), which means there is

9

Figure 10: Block diagram of the feedback loop with PI controller

enough phase margin away from instability point (−180°). On the otherhand, the phase margin will be less and less if integral gain continuesincreasing above 2⇡f

hbw

. It can be seen clearly in the bode diagram ofopen loop as shown in Figure 11. What’s worse is that there exists delayin the loop, which makes the feedback control with large integral gainvulnerable to causing instability. Figure 12 shows the case with 2µs delay,in which the instability arises inevitably as the integral gain becomeslarger and larger. In practice, the integral gain is set to 2⇡f

hbw

so as tokeep a constant loop phase and larger phase margin at all frequencies.

Having set the proper proportional and integral gains, it is able tolook into the closed loop performance against the noise. The proportionalgains of Kp=6 for normal conducting cavity and 50 for superconductingcavity, and the integral gain of K

i

= 2⇡fhbw

are firstly taken as analyzedabove. The noise suppressions in closed loops as a function of frequencyare given in Figures 13 and 14 for superconducting cavity and normalconducting cavity separately.

It is found from Figures 13 and 14 that with integral controller thefeedback loop can suppress e↵ectively the low frequency noise but the per-formance degrades as frequency increases, while the far higher frequencynoise is filtered by cavity itself. Assuming that 15° phase variation is in-duced by per 1% change of the cathode voltage, and 0.5° phase variationis to be achieved under feedback control, we can obtain the noise toler-ance as listed in Tables 3 and 4. For the modulator ripple, the troublefrequencies are usually less than hundred kHz. It is therefore valuablefor us to focus on the rising part of the curves in figures above and thecorresponding results in tables.

Tables 3 and 4 indicate that for a ripple of 3% in cathode voltage, theripple frequency must be limited to less than several hundred Hz in orderto achieve the required phase variation for superconducting and normalconducting cavities, and for a 1% ripple, the frequency has to be less thana couple of kHz for normal conducting cavity. Superconducting cavityfeedback loop has much better performance against noise at higher fre-quency due to its intrinsic ability to achieve higher proportional gain.As there exists other factors a↵ecting the feedback performance such as

10

!100

!50

0

50

100

150

200

Ma

gn

itu

de

(d

B)

10!1

100

101

102

103

104

105

106

107

108

!180

!135

!90

!45

0

Ph

as

e (

de

g)

Bode Diagram

Frequency (Hz)

Ki=0

Ki=0.001K

Ki=0.01K

Ki=0.1K

Ki=K

Ki=10K

Ki=100K

Ki=1000K

Ki<2pi*fhbw

Ki=2pi*fhbw

Ki<2pi*fhbw

Figure 11: Phase margin reduced in open loop under di↵erent integral gains(without delay, K = 2⇡fhbw)

Bode Diagram

Frequency (Hz)10

010

110

210

310

410

510

610

7

!225

!180

!135

!90

!45

0

Ph

ase (

deg

)

!50

0

50

100

150

200

System: Ki=KGain Margin (dB): 6.38At frequency (Hz): 1.25e+05Closed Loop Stable? Yes

System: Ki=0.1KGain Margin (dB): 6.77At frequency (Hz): 1.3e+05Closed Loop Stable? Yes

System: Ki=10KGain Margin (dB): !7.25At frequency (Hz): 5.4e+04Closed Loop Stable? No

Mag

nit

ud

e (

dB

)

Ki=0Ki=0.001KKi=0.01KKi=0.1KKi=KKi=10KKi=100KKI=1000K

Figure 12: Phase margin reduced in open loop under di↵erent integral gains(2µs delay, K = 2⇡fhbw)

11

Bode Diagram

Frequency (Hz)10

110

210

310

410

510

610

7!90

!80

!70

!60

!50

!40

!30

System: TFrequency (Hz): 302Magnitude (dB): !40

System: TFrequency (Hz): 6.83e+04Magnitude (dB): !40

Ma

gn

itu

de

(d

B)

Figure 13: Noise suppression performance of PI feedback closed loop as a func-tion of frequency for superconducting cavity(Kp = 50,Ki = 2⇡ × 518)

Bode Diagram

Frequency (Hz)10

210

310

410

510

610

7!70

!60

!50

!40

!30

!20

!10







Mag

nit

ud

e (

dB

)

Figure 14: Noise suppression performance of PI feedback closed loop as a func-tion of frequency for normal conducting cavity(Kp = 6,Ki = 2⇡ × 104)

12

Table 3: Noise tolerances of PI feedback closed loop at di↵erent frequencies(Superconducting cavity, Kp = 50,Ki = 2⇡ × 518 )

Frequency range/kHz

Gainavailable

Tolerance inoutput phase/°

Tolerance incathode voltage

<0.3, or >68 >100 >50 >3.3%0.3 ∼ 68 50 ∼ 100 25 ∼ 50 1.7% ∼ 3.3%

Table 4: Noise tolerances of PI feedback closed loop at di↵erent frequencies(Normal conducting cavity, Kp = 6,Ki = 2⇡ × 104 )

Frequency range/kHz

Gainavailable



<0.6, or >97 >100 >50 >3.3%0.6 ∼ 2,280 ∼ 970 30 ∼ 100 15 ∼ 50 1% ∼ 3.3%2 ∼ 7.5,150 ∼ 280 10 ∼ 30 5 ∼ 15 0.33% ∼ 1%

7.5 ∼ 150 4.5 ∼ 10 2.25 ∼ 5 0.15% ∼ 0.33%

unpredicted noise, cavity detuning and passband modes, the performanceof PI feedback against noise might degrade. A much worse case of propor-tional gain of 20 for superconducting cavity and 1 for normal conductingcavity is shown in Figures 15, 16 and Tables 5, 6. In both superconduct-ing and normal conducting cavities, the frequency is limited to less than100 Hz for 3% ripple, while less than several hundred Hz is needed for 1%ripple. Normal conducting cavity feedback loop has a poorer performancefor the frequencies of higher than 1 kHz, at which the ripple should belimited to the order of 0.1%.

In considering both cases above, to control the phase variation within0.5°, it therefore seems better to keep the modulator ripple <1% at lowfrequency (<1 kHz), while<0.1% for normal conducting cavity and<0.5%for superconducting cavity at higher frequency (>1 kHz). For the ripplewith frequency below 100 Hz, it is able to leave the ripple at around 3%or even higher for much lower frequencies.

The modulator droop can be seen as very low frequency ripple oreven DC component noise in feedback loop, and therefore theoreticallythe droop could be larger than 3% in cathode voltage. However, largerdroop or ripple will consume more power and reduce phase adjustmentrange of the feedback loop. For example, around 8% more power and45 ° in phase (assuming 15° is induced by per 1% variation in cathodevoltage) are needed to compensate 3% droop in feedback loop, while 13%more power and 75° in phase are needed for 5% droop. For the purposeof power e�ciency and achievement of good feedback performance, it isnecessary to reduce the droop and ripple as much as possible.

13

Bode Diagram

Frequency (Hz)10

110

210

310

410

510

610

7!90

!80

!70

!60

!50

!40

!30

!20





Ma

gn

itu

de

(d

B)

Figure 15: Noise suppression performance of PI feedback closed loop as a func-tion of frequency for superconducting cavity(Kp = 20,Ki = 2⇡ × 518)

Bode Diagram

Frequency (Hz)10

210

310

410

510

610

7!55

!50

!45

!40

!35

!30

!25

!20

!15

!10

!5







Ma

gn

itu

de

(d

B)

Figure 16: Noise suppression performance of PI feedback closed loop as a func-tion of frequency for normal conducting cavity(Kp = 1,Ki = 2⇡ × 104)

14

Table 5: Noise tolerances of PI feedback closed loop at di↵erent frequencies(Superconducting cavity, Kp = 20,Ki = 2⇡ × 518)

Frequency range/kHz

Gainavailable



<0.1, or >58 >100 >50 >3.3%0.1 ∼ 0.4,15 ∼ 58 30 ∼ 100 15 ∼ 50 1% ∼ 3.3%

0.4 ∼ 15 20 ∼ 30 10 ∼ 15 0.7% ∼ 1%

Table 6: Noise tolerances of PI feedback closed loop at di↵erent frequencies(Normal conducting cavity, Kp = 1,Ki = 2⇡ × 104)

Frequency range/kHz

Gainavailable



<0.1, or > 1000 >100 >50 >3.3%0.1˜0.3, 300˜1000 30 ∼ 100 15 ∼ 50 1% ∼ 3.3%0.3 ∼ 1,100 ∼ 300 10 ∼ 30 5 ∼ 15 0.33% ∼ 1%

1 ∼ 100 2 ∼ 10 1 ∼ 5 0.07% ∼ 0.33%

4 Conclusion

The modulator droop and ripple of per 1% induces more than 10° inklystron output phase and 1.25% in amplitude (2.5% in power). The PIfeedback loop has trouble to deal with the noises with high amplitude andhigh frequency due to loop delay and the necessity to keep proper phasemargin. It would be better to keep the modulator droop and ripple <1%in low frequency range (<1 kHz), while <0.1% for normal conductingcavity and <0.5% for superconducting cavity in higher frequency range(>1 kHz). For the modulator droop and very low frequency ripple below100 Hz, the tolerance for the ripple could be up to 3% or even higher, butit consumes more power and more phase dynamic range. It is essential toreduce the droop and ripple as much as possible.

References

[1] http://www.radartutorial.eu/08.transmitters/tx12.en.html.

[2] M. Hara, T. Nakamura, and T. Ohshima. A Ripple E↵ect of aKlystron Power Supply on Synchrotron Oscillation. Particle Accel-erators, 59(3):143–156, 1998.

[3] Jr. A.S.Gilmour. Klystrons, Traveling Wave Tubes, Magnetrons,Crossed-Field Amplifiers, and Gyrotrons, chapter Nonlinearities andDistrotion. Artech House, 2011.

15

[4] T. Kobayashi, E. Chishiro, T. Hori, H. Suzuki, M. Yamazaki,S. Anami, Z. Fang, Y. Fukui, M. Kawamura, and S.Michizono. Per-formance of J-PARC linac RF system. In Particle Accelerator Con-ference, 2007. PAC. IEEE, pages 2128–2130, 2007.

[5] M. McCarthy, M. Cro↵ord, and S. K ORNL. Spallation neutronsource superconducting linac klystron to cavity mismatch e↵ects andcompensation. In Proceedings of the 2008 Linear Accelerator Confer-ence, 2008.

[6] B. Chase and M. Champion. SNS LLRF Design Experience and itsPossible Adoption for the ILC. International Linear Collider Work-shop, 2005.

[7] P. Corredoura, R. Claus, L. Sapozhnikov, H. Schwarz, R. Tighe, andC. Ziomek. Low level RF system design for the PEP-II b factory. InParticle Accelerator Conference, 2002, volume 4, pages 2672–2674,2002.

[8] S. Chel, M. Desmons, A. Hamdi, and F. Peauger. New 1 MW 704MHz RF test stand at CEA-Saclay. Proc. EPAC08, 2008.

[9] M. Ho↵mann. Development of a multichannel RF field detector forthe Low-Level RF control of the Free-Electron Laser at Hamburg. PhDthesis, Hamburg University of Technology, 2008.

[10] T. Schilcher. Vector Sum Control of Pulsed Accelerating Fields inLorentz Force Detuned Superconducting Cavities. PhD thesis, Uni-versity of Hamburg, 1998.

[11] Hengjie Ma, Mark Champion, Mark Cro↵ord, Kay-Uwe Kasemir,Maurice Piller, Lawrence Doolittle, and Alex Ratti. Low-level rfcontrol of spallation neutron source: System and characterization.Phys. Rev. ST Accel. Beams, 9(3):032001, Mar 2006.

[12] S. Michizono, S. Anami, M. Kawamura, S. Yamaguchi, andT. Kobayashi. Digital rf control system for 400-mev proton linac ofjaeri/kek joint project. In Proceedings of the 2002 Linear AcceleratorConference, 2002.

[13] V. Ayvazyan, S. Choroba, Z. Geng, G. Petrosyan, S. Simrock, andD. Vladimir Vogel. OPTIMIZATION OF FILLING PROCEDUREFOR TESLA-TYPE CAVITIES FOR KLYSTRON RF POWERMINIMIZATION FOR EUROPEAN XFEL. In Proceedings of 1stInternational Particle Accelerator Conference: IPAC’10, 2010.

[14] Sung il Kwon, Joseph Bradley, Amy Regan, Yi-Ming Wang, andTony Rohlev. SNS MODELING - HIGH VOLTAGE POWER SUP-PLY RIPPLE ANALYSIS. Technical report, Los Alamos NationalLaboratory, 2000.

16

A Phase variation due to cathode volt-

age change in relativistic case

(� − 1)mc2 = eV (21)

� = eV

mc2+ 1 (22)

� =�

1 − 1

�2(23)

t = L

�= L

c�(24)

✓ = !t = !L

c�(25)

d✓

d�= − !L

c�2(26)

d�

d�= 1

2�

1 − 1�

2

⋅ 2�3= 1

��3(27)

d�

dV= e

mc2(28)

d✓

dV= d✓

d�

d�

d�

d�

dV= − e!L

mc3�3�3(29)

d✓ = − e!LV

mc3�3�3⋅ dVV

(30)

B Superconducting cavity feedback per-

formance with low proportional gain

17

Bode Diagram

Frequency (Hz)10

110

210

310

410

510

610

7!80

!70

!60

!50

!40

!30

!20





Mag

nit

ud

e (

dB

)

Figure 17: Noise suppression by feedback closed loop(Kp = 10,Ki = 2⇡ × 518)

Bode Diagram

Frequency (Hz)10

110

210

310

410

510

6!70

!60

!50

!40

!30

!20

!10

0System: TFrequency (Hz): 520Magnitude (dB): !6

System: TFrequency (Hz): 52.6Magnitude (dB): !20






Ma

gn

itu

de

(d

B)

Figure 18: Noise suppression by PI feedback closed loop(Kp = 1,Ki = 2⇡ × 518)

18

accelerator division - lunds universitet...delay/us hbw/hz critical gain hbw/hz critical gain 1 518...

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